Study of multi-dimensional problems arising in wave propagation using a hybrid scheme

Many scientific phenomena are linked to wave problems. This paper presents an effective and suitable technique for generating approximation solutions to multi-dimensional problems associated with wave propagation. We adopt a new iterative strategy to reduce the numerical work with minimum time efficiency compared to existing techniques such as the variational iteration method (VIM) and homotopy analysis method (HAM) have some limitations and constraints within the development of recurrence relation. To overcome this drawback, we present a Sawi integral transform (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}$$\end{document}ST) for constructing a suitable recurrence relation. This recurrence relation is solved to determine the coefficients of the homotopy perturbation strategy (HPS) that leads to the convergence series of the precise solution. This strategy derives the results in algebraic form that are independent of any discretization. To demonstrate the performance of this scheme, several mathematical frameworks and visual depictions are shown.

The current study aims to use a new iterative technique for multi-dimension challenges by combining S T and HPS.In the present work, we eliminate these drawbacks and constraints by offering a novel iterative method for these multi-dimensional wave issues.An iteration series with approximate findings that are close to the precise outcomes is produced by this new strategy.This technique performs more effectively and produces more appealing outcomes for the present challenges.The following is a description of this work: the concept of Sawi integral transform is given in "Fundamental concepts".In "Formulation of new iterative strategy", we build our new strategy to achieve the multi-dimension model findings.The convergence theorem has been laid out in "Convergence of new iterative strategy".In "Numerical applications", a few numerical examples are examined to demonstrate the power of new technique and we offer the conclusion at the end of "Conclusion remarks and future work".

Fundamental concepts
In this portion, we give few fundamental features of S T that are helpful in the development of our new strategy.

Sawi transform
Definition 2.1 Let ϑ be a function of η ≥ 0 .Then, S T is 26,27 in which S represents the symbol of S T. Now where Q(θ) shows the function of ϑ(η) .The S T of ϑ(η) for η ≥ 0 exist if ϑ(η) tends to exponentially ordered and piecewise continuous.The existence of S T for ϑ(η) is basically predicated on the two requirements mentioned.

Formulation of new iterative strategy
This section examines the approximate solutions of 1D, 2D, and 3D wave problems by using new iterative strategy (NIS).This approach can be used to solve differential equations based on initial conditions.We stated that the construction of this approach does not depend on integrating and other suppositions.Let a differential equation like that subjected to initial conditions where f (ϑ) denotes the nonlinear element, f (x 1 , η) is known component of arbitrary constants a 1 and a 2 , and ϑ(x 1 , η) is a uniform function.Moreover, we may express Eq.(4) like this: A function of a real variable can be transformed into an expression of a complex variable using an integral transformation known as the Sawi transform in mathematics.This transformation has several uses in the fields of science and technology because it serves as a tool to deal with differential problems.
Apply S T on Eq. ( 6), we get Using the formula as defined in Eq. ( 3), it yields Thus, Q(θ) is derived as On inverse S T on Eq. ( 7), we get Use the condition (5), we obtain This Eq. ( 8) is known as the development of NIS of Eq. ( 4).
Let HPS be introduced as where as the nonlinear variable f (ϑ) is stated as Hence, we are able to generate H ′ n s polynomial as Use Eqs. ( 9)- (11) in Eq. ( 8) and evaluate the similar components of p, it yields Following this procedure, which results in Hence, Eq. ( 12) provides a closed-form approximation to the differential problem.

Convergence of new iterative strategy
Proof Taking the series {F r } as a partial sum of Eq. ( 12), we obtain Vol:.( 1234567890 r=0 in B is a Cauchy sequence.It follows that the series solution of Eq. ( 12) is convergent.
represents the approximate series solution of Eq. (4), then maximal absolute error can be determined by in which δ is a digit which means �ϑ i+1 � �ϑ i � ≤ δ.

Numerical applications
We provide some numerical tests for showing the precision and reliability of NIS.We can observe that, as compared to other approaches, this method is substantially easier to apply in obtaining the convergence series.We illustrate the physical nature of the resulting plot distribution with graphical structures.Furthermore, a visual depiction of the error distribution demonstrated the near correspondence between the NIS outcomes and the precise results.We can compute the absolute error estimates by evaluating the exact solutions with the NIS values. (13) Vol Using the formula as defined in Eq. ( 3), it yields Thus, Q(θ) reveals as On inverse S T, we have Thus HPS yields such as By assessing comparable components of p, we arrive at Likewise, we can consider the approximation series in such a way that which can approaches to Figure 1 shows periodic soliton waves in two diagrams: Fig. 1a 3D surface plot for analytical results of ϑ(x 1 , η) and Fig. 1b shows 3D surface plot for precise results of ϑ(x 1 , η) for one-dimensional wave equation at −10 ≤ x 1 ≤ 10 and 0 ≤ η ≤ 0.01 .The effective agreement among analytical and the precise results at 0 ≤ x 1 ≤ 5 along η = 0.1 is shown in Fig. 2, which further validates the strong agreement of NIS for example (5.1).We can precisely propagate any surface to reflect the pertinent natural physical processes, according to this technique.The error p 0 : ϑ 0 (x 1 , η) = ϑ(x 1 , 0) = 2η cos(x 1 ), 5 5! cos(x 1 ), Table 1.Error distribution of ϑ(x 1 , η) along x 1 -space at different values.www.nature.com/scientificreports/distribution among analytical and precise results for ϑ(x 1 , η) along x 1 -space at different values is shown in Table 1.This contraction demonstrates the effectiveness of proposed technique in finding the closed-form results for the wave problems.

Example 2
Consider the two-dimensional wave equation subjected to initial and boundary conditions Apply S T on Eq. ( 27), we get Using the formula as defined in Eq. ( 3), it yields Thus, Q(θ) reveals as On inverse S T, we have Thus HPS yields such as By assessing comparable components of p, we arrive at Likewise, we can consider the approximation series in such a way that ) sin(y 1 ), 3 3! sin(x 1 ) sin(y 1 ), 5 5! sin(x 1 ) sin(y 1 ), 7 7! sin(x 1 ) sin(y 1 ), www.nature.com/scientificreports/which can approaches to Figure 3 shows periodic soliton waves in two diagrams: Fig. 3a: 3D surface plot for analytical results and Fig. 3b: 3D surface plot for precise results of ϑ(x 1 , y 1 , η) for two-dimensional wave equation at −5 ≤ x 1 ≤ 5 , 0 ≤ η ≤ 0.01 along y 1 = 0.5 .The effective agreement among analytical and the precise results at 0 ≤ x 1 ≤ 5 , y 1 = 0.1 along η = 0.1 is shown in Fig. 4, which further validates the strong agreement of NIS for example (5.2).We can pre- cisely propagate any surface to reflect the pertinent natural physical processes, according to this technique.The error distribution among analytical and precise results for ϑ(x 1 , y 1 , η) along x 1 -space at different values is shown in Table 2.This contraction demonstrates the effectiveness of proposed technique in finding the closed-form results for the wave problems.

Conclusion remarks and future work
In this article, we successfully applied the new iterative strategy for the approximate results of multi-dimensional wave problems.This technique uses the recurrence relation to produce the findings of the analysis.The findings obtained from numerical examples show that our technique is simple to implement and has a greater rate of convergence than existing approaches.The Sawi integral transform has the ability to control the global error, which makes it a suitable method for solving problems with rapidly changing solutions.The method is relatively easy to implement, especially for problems with periodic solutions.The 3D figures in the illustrated problems show the periodic soliton waves in the deep well.The physical behavior of the problems is depicted by the 3D graphical representations, and the visual inaccuracy between the exact outcomes and the produced results is represented by the 2D plot distribution.This method requires accurate initial guesses for the solution, which can be challenging in some cases.In terms of its effectiveness and efficiency, the Sawi integral transform is a relatively new method and has not been widely studied or compared to other numerical methods for solving PDEs.However, in the cases where it has been applied, it has shown promising results, with relatively high accuracy and efficiency compared to other methods.This composition of Sawi transform and the homotopy perturbation strategy gives the solution of multi-dimensional problems which is very useful in wave propagation.This novel iterative technique can also be used to solve other physical chemistry, engineering, and medical research challenges, such as calculating the growth rate of tumors, calculating the total quantity of infecting cells, calculating the amount of viral particles in blood during HIV-1 diseases, analyzing the impact of humidity on skew plate vibration, and calculating the amount of chemicals involved in chemical chain reactions in the future.

Figure 3 .
Figure 3. Surface results for two-dimensional problem.

Figure 4 .
Figure 4. Error between analytical and precise results.

Figure 5 .
Figure 5. Surface results for three-dimensional problem.

Figure 6 .
Figure 6.Error between analytical and precise results.